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In this paper the thresholds of the loss of Landau damping due to the presence of inductive impedance in a single and double harmonic RF systems are determined, both from calculations and particle simulations. High harmonic RF system, operating in bunch lengthening mode is used in many accelerators with space charge or inductive impedance to reduce the peak line density or stabilize the beam. An analytical approach, based on emerging of the discrete Van Kampen modes, shows that improved stability in a double RF system can be achieved only below some critical value of longitudinal emittance. Above this threshold, a phase shift of more than 15 degrees between the two RF components is proven to stabilize the bunch. These results, confirmed also by particle simulations, are able to explain now observations during the ppbar operation of the SPS. The thresholds in bunch shortening mode as well as in a single RF case are compared with this regime.
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